Matrix decompositions inspired by the DMP inverse and their spectral properties

This paper introduces a novel matrix decomposition associated with the DMP inverse, referred to as the DMP decomposition. A given square matrix A is decomposed as A=A1+A2, where A1 and A2 are defined through expressions involving the DMP inverse. According to properties of the DMP decomposition, the matrix A1 of index one retains all nonzero eigenvalues of A and their corresponding eigenvectors, whereas A2 retains the nilpotent structure of A. The properties of A1 and A2 are examined in detail, along with the relationships between the DMP and Drazin inverses of A1, A2, and A. Analysis of the DMP decomposition reveals a limitation: a 0-eigenvector of A does not necessarily correspond to a 0-eigenvector of A2. The DMP-D decomposition is introduced to address this limitation and further clarifies the behavior of eigenvectors associated with the zero eigenvalue. The DMP-D decomposition takes the form A=B1+B2+B3, where B1=A1. A central aspect is the further decomposition of A2 into two mutually annihilating components. As a result, all zero eigenvalues and their associated eigenvectors of a singular matrix A are contained entirely within B2. Furthermore, all zero eigenvalues and their associated eigenvectors of a singular matrix A are contained entirely within A2 iff rank(A1)+rank(A2)=rank(A).